find $f(a), f(a + h)$, and the difference quotient $\frac{f(a + h)-f(a)}{h}$, where $h\neq0$…

Explanation:

Step1: Substitute (f(a + h)) and (f(a)) into the difference - quotient formula

We know (f(a + h)=\frac{3a + 3h}{a + h-8}) and (f(a)=\frac{3a}{a - 8}). The difference quotient (\frac{f(a + h)-f(a)}{h}=\frac{\frac{3a + 3h}{a + h-8}-\frac{3a}{a - 8}}{h}).

Step2: Find a common denominator for the numerator

The common denominator of (a + h-8) and (a - 8) is ((a + h-8)(a - 8)). So (\frac{3a + 3h}{a + h-8}-\frac{3a}{a - 8}=\frac{(3a + 3h)(a - 8)-3a(a + h-8)}{(a + h-8)(a - 8)}). Expand the numerator: ((3a + 3h)(a - 8)=3a^{2}-24a+3ah - 24h) and (3a(a + h-8)=3a^{2}+3ah-24a). Then ((3a + 3h)(a - 8)-3a(a + h-8)=3a^{2}-24a+3ah - 24h-(3a^{2}+3ah-24a)=-24h). So (\frac{(3a + 3h)(a - 8)-3a(a + h-8)}{(a + h-8)(a - 8)}=\frac{-24h}{(a + h-8)(a - 8)}).

Step3: Divide by (h)

(\frac{\frac{-24h}{(a + h-8)(a - 8)}}{h}=\frac{-24}{(a + h-8)(a - 8)}).

Answer:

(\frac{-24}{(a + h-8)(a - 8)})